用于求解多域瞬态偏微分方程的局部神经算子
Local neural operator for solving transient partial differential equations on varied domains
AI总结：
针对神经网络在固定域训练导致求解多域瞬态PDEs受限的问题，提出局部神经算子（LNO）及边界处理策略，预训练模型可求解未见过的域，在翼型叶栅流等问题上比传统方法快约1000倍，大幅提升效率。
摘要：
人工智能（AI）在降低求解偏微分方程（PDEs）的巨大成本方面展现出巨大潜力。然而，由于神经网络在固定域和边界上定义和训练，其在实践中尚未完全实现这一潜力。本文提出局部神经算子（LNO）以求解多域瞬态PDEs，并结合边界处理等便捷策略，使一个预训练LNO能预测不同域上的解。为验证效果，LNO从随机生成的数据样本中学习Navier-Stokes方程，预训练后的LNO被用作显式数值时间推进格式，求解未见过的域上的流体流动问题（如驱动腔流和翼型叶栅流）。在翼型叶栅流求解中，其速度比传统有限元方法快约1000倍。预训练LNO的求解过程效率极高，在实际中加速数值计算的潜力显著。
英文摘要：
Artificial intelligence (AI) shows great potential to reduce the huge cost of solving partial differential equations (PDEs). However, it is not fully realized in practice as neural networks are defined and trained on fixed domains and boundaries. Herein, we propose local neural operator (LNO) for solving transient PDEs on varied domains. It comes together with a handy strategy including boundary treatments, enabling one pre-trained LNO to predict solutions on different domains. For demonstration, LNO learns Navier-Stokes equations from randomly generated data samples, and then the pre-trained LNO is used as an explicit numerical time-marching scheme to solve the flow of fluid on unseen domains, e.g., the flow in a lid-driven cavity and the flow across the cascade of airfoils. It is about 1000$\times$ faster than the conventional finite element method to calculate the flow across the cascade of airfoils. The solving process with pre-trained LNO achieves great efficiency, with significant potential to accelerate numerical calculations in practice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
